A Discrete-time Valuation of Callable Financial Securities with Regime
Switches
Kimitoshi Sato
1
and Katsushige Sawaki
2
1
Graduate School of Finance, Accounting and Law, Waseda University,
1-4-1, Nihombashi, Chuo-ku, Tokyo, 103-0027, Japan
2
Graduate School of Business Administration, Nanzan University, 18 Yamazato-cho, Showa-ku, Nagoya, 466-8673, Japan
Keywords:
Optimal Stopping, Game Option, Markov Chain, Regime Switching, Callable Securities, Stopping Bound-
aries.
Abstract:
In this paper, we consider a model of valuing callable financial securities when the underlying asset price
dynamic is modeled by a regime switching process. The callable securities enable both an issuer and an
investor to exercise their rights to call. We show that such a model can be formulated as a coupled stochastic
game for the optimal stopping problem with two stopping boundaries. We provide analytical results of optimal
stopping rules of the issuer and the investor under general payoff functions defined on the underlying asset
price, the state of the economy and the time. In particular, we derive specific stopping boundaries for the both
players by specifying for the callable securities to be the callable American put option.
1 INTRODUCTION
The purpose of this paper is to develop a dynamic
valuation framework for callable financial securities
with general payoff function by explicitly incorporat-
ing the use of regime switches. Such examples of
the callable financial security may include game op-
tions (Kifer, 2000), (Kyprianou, 2004), convertible
bond (Yagi and Sawaki, 2005), (Yagi and Sawaki,
2007), callable put and call options (Black and Sc-
holes, 1973), (Brennan and Schwartz, 1976), (Geske
and Johnson, 1984). Most studies on these securities
have focused on the pricing of the derivatives when
the underlying asset price processes follow a Brown-
ian motion defined on a single probability space. In
other words the realizations of the price process come
from the same source of the uncertainty over the plan-
ning horizon.
The Markov regime switching model make it pos-
sible to capture the structural changes of the under-
lying asset prices based on the macro-economic en-
vironment, fundamentals of the real economy and fi-
nancial policies including international monetary co-
operation. Such regime switching can be presented
by the transition of the states of the economy, which
follows a Markov chain. Recently, there is a grow-
ing interest in the regime switching model. (Naik,
1993), (Guo, 2001), (Elliott et al., 2005) address the
European call option price formula. (Guo and Zhang,
2004) presents a valuation model for perpetual Amer-
ican put options. (Le and Wang, 2010) study the op-
timal stopping time for the finite time horizon, and
derive the optimal stopping strategy and properties of
the solution. They also derive the technique for com-
puting the solution and show some numerical exam-
ples for the American put option.
In this paper we show that there exists a pair of op-
timal stopping rules for the issuer and of the investor
and derive the value of the coupled game. Should the
payoff functions be specified like options, some an-
alytical properties of the optimal stopping rules and
their values can be explored under the several assump-
tions. In particular, we are interested in the cases of
callable American put option in which we may derive
the optimal stopping boundaries of the both of the is-
suer and the investor, depending on the state of the
economy. Numerical examples are also presented to
illustrate these properties.
The organizationof our paper is as follows: In sec-
tion 2, we formulate a discrete time valuation model
for a callable contingent claim whose payoff func-
tions are in general form. And then we derive opti-
mal policies and investigate their analytical properties
by using contraction mappings. Section 3 discusses
a case of the payoff functions to derive the specific
stop and continue regions for callable put. In Sec-
77
Sato K. and Sawaki K..
A Discrete Time Valuation of Callable Financial Securities with Regime Switches.
DOI: 10.5220/0004201402250229
In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 225-229
ISBN: 978-989-8565-40-2
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
tion 4 we present numerical results for the American
callable put option using binomial model. Finally, last
section concludes the paper with further comments. It
summarize results of this paper and raises further di-
rections for future research.
2 A GENETIC MODEL OF
CALLABLE-PUTABLE
FINANCIAL COMMODITIES
In this section we formulate the valuation of callable
securities as an optimal stopping problem in discrete
time. Let T be the time index set {0, 1,···}. We con-
sider a complete probability space (,F ,P ), where
P is a real-world probability. We suppose that the
uncertainties of an asset price depend on its fluctu-
ation and the economic states which are described
by the probability space (,F ,P ). Let {1,2,··· ,N}
be the set of states of the economy and i or j de-
note one of these states. We denote Z := {Z
t
}
tT
be the finite Markov chain with transition probabil-
ity P
ij
= Pr{Z
t+1
= j | Z
t
= i}. A transition from i to
j means a regime switch. Let r be the market interest
rate of the bank account. We suppose that the price
dynamics B := {B
t
}
tT
of the bank account is given
by
B
t
= B
t1
e
r
, B
0
= 1.
Let S := {S
t
}
tT
be the asset price at time t. We sup-
pose that {X
i
t
} be a sequence of i.i.d. random variable
having mean µ
i
with the probability distribution F
i
(·)
and its parameters depend on the state of the economy
modeled by Z. Here, the sequence {X
i
t
} and {Z
t
} are
assumed to be independent. Then, the asset price is
defined as
S
t+1
= S
t
X
i
t
. (1)
The Esscher transform is well-known tool to de-
termine an equivalent martingale measure for the val-
uation of options in an incomplete market ((Elliott
et al., 2005) and (Ching et al., 2007)). (Ching et al.,
2007) define the regime-swiching Esscher transform
in discrete time and apply it to determine an equiva-
lent martingale measure when the price dynamics is
modeled by high-order Markov chain.
We define Y
i
t
= logX
i
t
and Y := {Y
t
}
tT
. Let F
Z
t
and F
Y
t
denote the σ-algebras generated by the val-
ues of Z and Y, respectively. We set G = F
Z
t
F
Y
t
for t T . We assume that θ
t
be a F
Z
T
-measurable
random variable for each t = 1,2,···. It is interpreted
as the regime-switching Esscher parameter at time t
conditional on F
Z
T
. Let M
Y
(t,θ
t
) denote the moment
generating function of Y
i
t
given F
Z
T
under P , that is,
M
Y
(t,θ
t
) := E[e
θ
t
Y
i
t
| F
Z
T
]. We define P
θ
as a equiva-
lent martingale measure for P on G
T
associated with
(θ
1
,θ
2
,··· ,θ
T
).
The next proposition follows from (Ching et al.,
2007).
Proposition 2.1. The discounted price process
{S
t
/B
t
}
tT
is a (G , P
θ
)-martingale if and only if θ
t
satisfies
M
Y
(t + 1,θ
t+1
+ 1)
M
Y
(t + 1,θ
t+1
)
= e
r
. (2)
If the dynamics Y is governed by the following
Markov-modulated binomial model:
P(Y
i
t
= y) =
(
p(Z
t
), if y = b(Z
t
),
1 p(Z
t
), if y = a(Z
t
),
(3)
then the following proposition provides the Esscher
transform of this process. For simplicity of notation,
we write p
t
, a
t
and b
t
instead of p(Z
t
), a(Z
t
) and
b(Z
t
), respectively.
Proposition 2.2. The Esscer transform of the Markov
modulated binomial model with parameter p
t
is again
a binomial model with the parameter
e
r
e
a
t
e
b
t
e
a
t
.
A callable contingent claim is a contract between
an issuer I and an investor II addressing the asset with
a maturity T. The issuer can choose a stopping time
σ to call back the claim with the payoff function f
σ
and the investor can also choose a stopping time τ to
exercise his/her right with the payoff function g
τ
at
any time before the maturity. Should neither of them
stop before the maturity, the payoff is h
T
. The payoff
always goes from the issuer to the investor. Here, we
assume
0 g
t
h
t
f
t
, 0 t < T
and
g
T
= h
T
. (4)
The investor wishes to exercise the right to maximize
the expected payoff. On the other hand, the issuer
wants to call the contract to minimize the payment to
the investor. Then, for any pair of the stopping times
(σ,τ), define the payoff function by
R(σ,τ) = f
σ
1
{σ<τT}
+ g
τ
1
{τ<σT}
+ h
T
1
{στ=T}
.
(5)
When the initial asset price S
0
= s, our stopping prob-
lem becomes the valuation of
v
0
(s,i) = min
σJ
0,T
max
τJ
0,T
E
θ
s,i
[β
στ
R(σ,τ)], (6)
where β e
r
, 0 < β < 1 is the discount factor, J
is the finite set of stopping times taking values in
ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems
78
{0,1,··· ,T}, and E
θ
[·] is an expectation under P
θ
.
Since the asset price process follows a random walk,
the payoff processes of g
t
and f
t
are both Markov
types. We consider this optimal stopping problem as
a Markov decision process. Let v
n
(s,i) be the price
of the callable contingent claim when the asset price
is s and the state is i. Here, the trading period moves
backward in time indexed by n = 0, 1,2,· ·· ,T. It is
easy to see that v
n
(s,i) satisfies
v
n+1
(s,i) (U v
n
)(s,i)
min{ f
n+1
(s,i),max(g
n+1
(s,i),A v
n
)}
(7)
with the boundary conditions are v
0
(s,i) = h
0
(s,i) for
any s, i and v
n
(s,0) 0 for any n and s. A is the
operator defined by
(A v
n
)(s,i) β
N
j=1
P
ij
Z
0
v
n
(sx, j)dF
i
(x). (8)
Remark 2.1. The equation (7) can be reduced to the
non-switching model when we set P
ii
= 1 for all i, or
f
n
(s,i) = f
n
(s), g
n
(s,i) = g
n
(s), h
0
(s,i) = h
0
(s) and
µ
i
= µ for all i, n and s.
Let V be the set of all bounded measurable func-
tions with the norm kvk = sup
s(0,)
|v(s,i)| for any
i. For u,v V, we write u v if u(s,i) v(s,i) for
all s (0,). A mapping U is called a contraction
mapping if
kU u U vk βku vk
for some β < 1 and for all u,v V.
Lemma 2.1. The mapping U as defined by equation
(7) is a contraction mapping.
Corollary 2.1. There exists a unique function v V
such that
(U v)(s,i) = v(s, i) for all s,i. (9)
Furthermore, for all u V,
(U
T
u)(s,i) v(s,i) as T ,
where v(s,i) is equal to the fixed point defined by
equation (9), that is, v(s,i) is a unique solution to
v(s,i) = min{ f(s,i),max(g(s,i),A v)}.
Since U is a contraction mapping from Corol-
lary 2.1, the optimal value function v for the perpetual
contingent claim can be obtained as the limit by suc-
cessively applying an operator U to any initial value
function v for a finite lived contingent claim.
To establish an optimal policy, we make some as-
sumptions;
Assumption 2.1.
(i) F
1
(x) F
2
(x) ·· · F
N
(x) for all x.
(ii) f
n
(s,i) f
n
(s, j), g
n
(s,i) g
n
(s, j) and
h
n
(s,i) h
n
(s, j) for each n and s, and states i,
j, 1 j < i N.
(iii) f
n
(s,i), g
n
(s,i) and h
n
(s,i) are monotone in s for
each i and n, and are non-decreasing in n for
each s and i.
(iv) For each k N,
N
j=k
P
ij
is non-decreasing in i.
Assumption (i) means X
i+1
n
first order stochasti-
cally dominates X
i
n
for any i and n. That is, as the
state i increases, the economy is going well. Thus,
the state N represents that the most ”Good” econ-
omy. Assumption (ii) implies that the payoff values
increase as the economy is getting better. In addition,
by Assumption (iii), the payoff values decreases as
the maturity approaches. Assumption (iv) asserts that
the probability of a transition into any block of states
{k, k + 1,·· ·} is an increasing function of the present
state.
Lemma 2.2. Suppose Assumption 2.1 holds.
(i) For each i, (U
n
v)(s,i) is monotone in s for v V.
(ii) v satisfying v = U v is monotone in s.
(iii) Suppose v
n
(s,i) is monotone non-decreasing in
s, then v
n
(s,i) is non-decreasing in i.
(iv) v
n
(s,i) is non-decreasing in n for each s and i.
(v) For each i, there exists a pair (s
n
(i),s
∗∗
n
(i)),
s
∗∗
n
(i) < s
n
(i), of the optimal boundaries such
that
v
n
(s,i) (U v
n1
)(s)
=
f
n
(s,i), if s
n
(i) s,
A v
n1
, if s
∗∗
n
(i) < s < s
n
(i),
g
n
(s,i), if s s
∗∗
n
(i),
with v
0
(s,i) = h
0
(s,i).
Corollary 2.2. The relationship between g
n
, f
n
and
v
n
(s,i) is given by
g
n
(s,i) v
n
(s,i) f
n
(s,i).
We define the stopping regions S
I
for the issuer
and S
II
for the investor as
S
I
n
(i) = {(s,n, i) | v
n
(s,i) f
n
(s,i)}, (10)
S
II
n
(i) = {(s,n, i) | v
n
(s,i) g
n
(s,i)}. (11)
Moreover, the optimal exercise boundaries for the is-
suer and the investor are defined as
s
n
(i) = inf{s S
I
n
(i)}, (12)
s
∗∗
n
(i) = inf{s S
II
n
(i)}. (13)
ADiscreteTimeValuationofCallableFinancialSecuritieswithRegimeSwitches
79
3 A SIMPLE CALLABLE
AMERICAN PUT OPTION
WITH REGIME SWITCHING
Interesting results can be obtained for the special
cases when the payoff functions are specified. In
this section we consider callable American put option
whose payoff functions are specified as a special case
of callable contingent claim. If the issuer call back
the claim in period n, the issuer must pay to the in-
vestor g
n
(s,i) + δ
i
n
. Note that δ
i
n
is the compensate
for the contract cancellation, and varies depending on
the state and the time period. If the investor exercises
his/her right at any time before the maturity, the in-
vestor receives the amount g
n
(s,i). We discuss the
optimal cancel and exercise policies both for the is-
suer and investor and show the analytical properties
under some conditions.
We consider the case of a callable put op-
tion where g
n
(s,i) = max{K
i
s,0} and f
n
(s,i) =
g
n
(s,i) + δ
i
n
. The stopping regions for the issuer S
I
n
(i)
and the investor S
II
n
(i) with respect to the callable put
option are given by
S
I
n
(i) = {s | v
n
(s,i) (K
i
s)
+
+ δ
i
n
}, for n 1,
S
I
n
(i) = φ, for n = 0,
S
II
n
(i) = {s | v
n
(s,i) (K
i
s)
+
}, for n 0.
For each i and n, we define the optimal exercise
boundaries for the issuer ˜s
n
(i) and the investor ˜s
∗∗
n
(i)
as
˜s
n
(i) = inf{s | v
n
(s,i) = (K
i
s)
+
+ δ
i
n
},(14)
˜s
∗∗
n
(i) = inf{s | v
n
(s,i) = (K
i
s)
+
}. (15)
Assumption 3.1.
(i) βµ
N
1
(ii) 0 K
1
K
2
··· K
N
.
(iii) 0 δ
1
n
δ
2
n
··· δ
N
n
for each n.
(iv) δ
i
0
= 0 and δ
i
n
is non-decreasing and concave in
n > 0 for each i.
(v) β
N
j=1
P
ij
K
j
K
i
is non-decreasing in i.
Assumption (i) means the expected rate of vari-
ability for the asset price is less than or equal to
1
β
= e
r
. Assumption (ii) and (iii) imply that the strike
price and the compensate increase as the economy is
getting better. These assumptions consistent with the
Assumption 2.1 (ii). Assumption (iv) shows that the
compensate becomes smaller and smaller as the matu-
rity approaches. Assumption (v) asserts that the dif-
ference between the discounted expected value of a
strike price when the state transits to any state and the
strike price of present state is an non-decrasing func-
tion of the present state.
Theorem 3.1. Suppose that Assumption 3.1 (i)-(v)
holds. The stopping regions for the issuer and in-
vestor can be obtained as follows;
(i) The optimal stopping region for the issuer:
(
S
I
n
(i) = {K
i
}, if n
i
n T,
S
I
n
(i) = φ, if 0 n < n
i
,
(16)
where 0 K
1
K
2
··· K
N
, and n
i
inf{n |
δ
i
n
v
a
n
(K
i
,i)} which is non-decreasing in i.
Here, v
a
n
(s,i) = max{(K
i
s)
+
,A v
n1
(s,i)}.
(ii) The optimal stopping region for the investor:
(
S
II
n
(i) = [0, ˜s
∗∗
n
(i)], if n > 0,
S
II
0
(i) = {K
i
}, if n = 0,
(17)
where ˜s
∗∗
n
(i) is non-increasing in n and i. More-
over, ˜s
∗∗
n
(i) ˜s
n
(i) for each i and n.
4 NUMERICAL EXAMPLES
In this section we provide a numerical example for a
callable American put option by using the binomial
tree model. We assume that the transition probability
matrix is given by
P =
p
1
1 p
1
1 p
2
p
2
. (18)
For a fixed T, let us divide the interval [0,T] into M
subintervals such that T = hM. By Proposition 2.2,
the probability of upward in the state i is given by
q
i
=
e
rh
d
i
u
i
d
i
, i = 1,2, (19)
where u
i
= e
b
i
, d
i
= e
b
i
. Let u
i, j
and d
i, j
be the up-
ward and downward rate when the state changes from
i to j, respectively. The probability distribution func-
tion of X
i
t
is described by
P(X
i
t
= x) =
q
i
p
i
, if x = u
i,i
,
q
i
(1 p
i
), if x = u
i, j
,
(1 q
i
)p
i
, if x = d
i,i
,
(1 q
i
)(1 p
i
), if x = d
i, j
,
(20)
where i = 1, 2, i 6= j. It is easy to show that the process
is a martingale. The asset price after n periods on tree
can be obtained by
S
n
= S
0
u
n
1
0
u
n
2
1
d
n
3
0
d
n
4
1
(21)
where n
1
+ n
2
+ n
3
+ n
4
= n.
We set the parameters as T = 1, M = 300, r = 0.1,
b
1
= 0.03, b
2
= 0.01, p
1
= 0.7, p
2
= 0.8, K
1
= K
2
=
100, δ
i
n
= δ
1
= 5 and δ
2
n
= δ
2
= 6 for all n. These
parameters satisfy and Assumption 2.1 (i), (iv) and
Assumption 3.1. The optimal exercise regions for the
issuer and the investor is represented in Figure 1.
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80
70
75
80
85
90
95
100
105
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
˜s
n
(2)
˜s
n
(1)
˜s
∗∗
(1)
˜s
∗∗
(2)
n
2
n
1
Asset price
Time
Figure 1: Optimal exercise boundaries for the callable
American put.
5 CONCLUDING REMARKS
In this paper we consider the discrete time valuation
model for callable contingent claims in which the
asset price depends on a Markov environment pro-
cess. The model explicitly incorporates the use of
the regime switching. It is shown that such valuation
model with the Markov regime switches can be for-
mulated as a coupled optimal stopping problem of a
two person game between the issuer and the investor.
In particular, we show under some assumptions that
there exists a simple optimal call policy for the issuer
and optimal exercise policy for the investor which can
be described by the control limit values. If the distri-
butions of the state of the economy are stochastically
ordered, then we investigate analytical properties of
such optimal stopping rules for the issuer and the in-
vestor, respectively, possessing a monotone property.
ACKNOWLEDGEMENTS
This paper was supported in part by the Grant-in-Aid
for Scientific Research (No. 20241037) of the Japan
Society for the Promotion of Science in 2008-2012.
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